## Using Supporting Hyperplane Techniques in Solving Generalized Convex MINLP Problems

Tapio Westerlund, Ville-Pekka Eronen, Marko M. Mäkelä, Using Supporting Hyperplane Techniques in Solving Generalized Convex MINLP Problems. TUCS Technical Reports 1186, TUCS, 2017.

### Abstract:

Solution methods for convex mixed integer nonlinear programming (MINLP) problems have, usually, proven convergence properties if the functions involved are differentiable and convex. For other classes of convex MINLP problems fewer results have been given. Classical differential calculus can, though, be generalized to more general classes of functions than differentiable, via subdifferentials and subgradients. In addition, more general than convex functions can be included in a convex problem if the functions involved are defined from convex level sets, instead of being defined as convex functions only. The notion \textit{generalized convex}, used in the heading of this paper, refers to such additional properties.

The generalization for the differentiability is made by using subgradients of Clarke's subdifferential. Thus, all the functions in the problem are assumed to be locally Lipschitz continuous. The generalization of the functions is done by considering quasiconvex functions. Thus, instead of differentiable convex functions, nondifferentiable quasiconvex functions can be included in the actual problem formulation and a combined supporting hyperplane and cutting plane approach is given for the solution of the considered MINLP problem. Convergence to a global minimum is proved for the algorithm, when minimizing an $f^{\circ}$-pseudoconvex function, subject to $f^{\circ}$-pseudoconvex constraints. With some additional conditions, the proof is also valid for quasiconvex constraint functions, which sums up the properties of the method, treated in the paper.

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### BibTeX entry:

@TECHREPORT{tWeErMx17a,  title = {Using Supporting Hyperplane Techniques in Solving Generalized Convex MINLP Problems},  author = {Westerlund, Tapio and Eronen, Ville-Pekka and Mäkelä, Marko M.},  number = {1186},  series = {TUCS Technical Reports},  publisher = {TUCS},  year = {2017},  keywords = {Nonsmooth optimization; MINLP; Generalized convexities; Clarke generalized derivatives; Cutting planes; Supporting hyperplanes},  ISBN = {978-952-12-3578-8},}

Belongs to TUCS Research Unit(s): Turku Optimization Group (TOpGroup)

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